Abstract: Suppose is a proper PL map between PL manifolds and of dimension and respectively, compact. J. F. P. Hudson has shown that associated with each such map that is an embedding on is an element in when is odd and an element in when is even. These elements are invariant under a homotopy relative to . We show that, under slight additional assumptions on and is homotopic to an embedding if and only if for odd and for even. This result is used to give a sufficient condition for extending an embedding ( denotes -dimensional ball) to an embedding .

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